Yet Everton travel to Old Trafford on Wednesday above them in the league and playing with a style and confidence which suggests they are realistic top-four contenders.
When the new Premier League season kicked off in August all eyes were on Moyes as he began the almost impossible task of succeeding Britain's most successful football manager.
United's 12-match unbeaten streak after a dismal start has lifted the spotlight slightly away from Moyes and it is Spaniard Martinez who is suddenly attracting attention.
His side have lost only once in 13 league matches, are only below fourth-placed Liverpool on goal difference and thrashed Stoke City 4-0 at the weekend with a performance that showed the Goodison Park club have quickly adopted Martinez's philosophy.
A positive result at United on Wednesday and it would be hard to argue against the view that of all the changes in the managerial merry-go-round in the summer, Martinez's appointment by Everton has had the greatest impact so far.
Martinez, who led Wigan Athletic to FA Cup glory last season despite relegation, says his side must now start to believe they can really mix it with the big boys.
"This week will have a huge bearing on our season," Martinez, whose side then go to leaders Arsenal on Sunday, told a news conference this week.
"It would be very easy for me to play it down but we are going to go to Old Trafford and the Emirates.
"They are games that we are not expected to win but, clearly, if we want to achieve something special, we need to be able to break the mentality of going to the big grounds and getting some wins.
"Over the last 10 years, we have not beaten Arsenal, Manchester United, Chelsea and Liverpool at their grounds and we need to address that. We need to make sure we are ourselves," Martinez added.
"We need to know how we are going to play and who we are. We are excited about going to Old Trafford."
Martinez's Wigan side were noted for their brave attacking style, even if ultimately it could not prevent them from the drop. It was an approach that earned notable wins against Arsenal, Chelsea and Liverpool and Martinez says he will employ the same positive tactics with Everton.
"If you are going to be brave and try to win games rather than just being in a position where you want to keep a clean sheet and get through in damage-limitation situations, then you need to be flexible as a player and as a team and play in different ways in the modern game," Martinez said.
"That is the attitude we have had from the beginning. We have had one defeat in 13.
"The next test and the next challenge is can we go to the big places like the Old Trafford, the Emirates, Anfield and Stamford Bridge and get wins? That is not going to happen overnight. Probably on Wednesday will be the first step towards achieving that."
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